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Historia Mathematica 37 (2010) 428–459

www.elsevier.com/locate/yhmat

What Descartes knew of mathematics in 1628

David Rabouin *

CNRS-Univ. Paris Diderot, UFR “Science de la vie”, UMR 7219, Laboratoire de philosophie et d’Histoire des Sciences,Case 7093, 5 rue Thomas Mann, 75205 Paris Cedex 13, France

Abstract

The aim of this paper is to give an account of Descartes’ mathematical achievements in 1628–1629 using, asfar as is possible, only contemporary documents, and in particular Beeckman’s Journal for October 1628. In thefirst part of the paper, I study the content of these documents, bringing to light the mathematical weaknessesthey display. In the second part, I argue for the significance of these documents by comparing them with otherindependent sources, such as Descartes’ Regulae ad directionem ingenii. Finally, I outline the main consequencesof this study for understanding the mathematical development of Descartes before and after 1629.� 2010 Elsevier Inc. All rights reserved.

Résumé

Cet article se proposer d’étudier le développement mathématique de Descartes en 1628–1629 en restant auplus près des sources existantes. Le principal témoignage est donné par un passage du Journal de Beeckman datéd’Octobre 1628. Dans la première partie de l’article, j’analyse en détail ces documents en insistant sur lesnombreuses faiblesses qu’ils présentent. Dans la seconde partie, j’essaye de montrer quelles raisons nouspouvons avoir de prendre ces faiblesses au sérieux en les confrontant à des sources indépendantes, en particulierles Regulae ad directionem ingenii. J’essaye ensuite d’en tirer les conséquences quant à l’évaluation d’ensembledu développement mathématique de Descartes avant et après 1629.� 2010 Elsevier Inc. All rights reserved.

MSC code: 01A45

Keywords: Descartes; Beeckman; Regulae

0315-0860/$ - see front matter � 2010 Elsevier Inc. All rights reserved.

doi:10.1016/j.hm.2010.04.002

* Fax: +33 1 57 27 63 29.E-mail address: [email protected]

http://dx.doi.org/10.1016/j.hm.2010.04.002

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What Descartes knew in 1628 429429

1. Introduction

The starting point of this paper is a puzzling testimony given in his Journal1 by IsaacBeeckman, a Dutch scholar whom Descartes met in 1618 and considered “the promoterof his studies” [AT I, 161]. In October 1628, the French philosopher came to Dordrechtto visit his old friend, for whom he had written the Compendium musicae ten years beforeand to whom he presented his project of reforming all sciences, not long before leaving Hol-land in 1619. He described to Beeckman the results of his past nine years of study, especiallyin mathematics, during which, he said “he had made as much progress as was possible for ahuman mind” [Beeckman, 1939–1953, Vol. III, p. 94, fol. 333r]. As proof, he gave Beeckmana specimen of a “general algebra.” He also promised to send a treatise on algebra, by whichhe claimed one could bring geometry to perfection and, more than that, all human knowl-edge [Beeckman, 1939–1953, Vol. III, p. 95, fol. 333r]. But when one looks at the specimenthat was transcribed by Beeckman under the title Algebrae Des Cartes specimen quoddamand which I will study in Section 2.3, one is struck by its very poor and apparently confusedmathematical content.

The discovery of these documents at the beginning of the 20th century has not discour-aged commentators from proposing many reconstructions of Descartes’ philosophical andmathematical development in which he is supposed to have reached a high level oftechnique before 1628, sometimes as early as 1619–1620. What I intend in this article is abit of deflationist history. As a methodological exercise, I shall stick as far as possible tothe texts to see if it is possible to build a coherent interpretation of them without makingassumptions for which we have no evidence. Hence my first task will be to provide reasonswe might have for taking Beeckman’s testimony seriously, and thus consider that in 1628Descartes was not mathematically well advanced in the topics that he discussed with hisfriend. This implies the necessity of looking for confirmation in other sources, and this iswhat I shall do in the second part of the paper. The main document in this regard is theRegulae ad directionem ingenii,2 in which Descartes presented a schematism (i.e., a spatialrepresentation of algebraic operations) very close to that presented to Beeckman in 1628,and which seems to have been written, at least in part, during the same period.3 As I will

1 The manuscript of Beeckman’s Journal was recovered by Cornelis de Waard in the Provinciallibrary of Zeeland in 1905, so Charles Adam could insert the passages related to Descartes inVolume X of Les Oeuvres de Descartes (1908), which he launched with Paul Tannery before thedeath of the latter in 1904. See [Descartes, 1964–1974, pp. 17–78, 151–169 and 331–348. From nowon, AT]. De Waard gave an edition of it in four volumes [Beeckman, 1939–1953].

2 The Rules for the direction of the mind is an unfinished methodological treatise, which was notpublished and does not contain a clear indication of date [AT X, 359–469, from now on Regulae].It was written in Latin and was left by Descartes in his papers at his death. It consists of a seriesof Rules, of which only the first 21 are extant, Rules XIX–XXI subsisting only through theirplanned titles. The original manuscript is now lost and all existing editions were made from copies.The first one was a translation in Dutch in 1684, followed by a Latin edition published in RenatiDescartes Opuscula posthuma [Descartes, 1701]. For the tormented history of the treatise, see[Descartes, 1966].

3 One of the main arguments is the fact that Descartes mentions, in Rule VIII, the study of the“anaclastic curve” and concludes, “Even though the anaclastic has been the object of much fruitless

430 D. Rabouin

try to show, the mathematical weaknesses in these two projects are strikingly similar. Morethan that, these weaknesses, which seem incompatible with the techniques employed inLa Géométrie, are fully compatible with those employed in an early Cartesian treatise, oftenforgotten by commentators: The Progymnasmata de solidorum elementis.4

This strategy also requires giving a new account of the relationship between the pas-sages in Beeckman’s Journal of October 1628 and the text Descartes sent Beeckman afew months later and which Descartes described as his “most outstanding discovery”:the “construction” of the third and fourth degree equation by the intersection of a para-bola and a circle — a technique much more sophisticated than anything that could beexpected from the “sample” presented in 1628.5 I emphasize that these various documentswere presented separately to Beeckman, as were the other works done on curves (conicsections), which were produced at that time by Descartes in the context of optics andthrough purely geometrical means. There is no question of studying curves through alge-braic techniques in the documents produced in 1628–1629, and the program presentedto Beeckman is not that of a new classification of curves. How are we to understand thisfact if this classification of curves is supposed to have been at the core of Descartes’ pro-gram since 1619?

My claim is that one should resist the temptation of projecting onto this set of docu-ments the unified view permitted by the “geometrical calculus” presented in 1637 inLa Géométrie.6 In fact, this methodological requisite seems of importance if one wants tounderstand the breakthrough leading to the very idea of this “calcul géométrique,” thatis, how what were separated in 1628–1629 (the “general algebra,” the “construction” ofequations, and the study of curves) came to be gathered in a unified treatment. As I willtry to show, it is not sufficient to paste together the different attempts presented in 1628to regain a unified treatment, and there were deep reasons for these elements to remainseparated at that time.

research in the past, I can see nothing to prevent anyone who uses our method exactly fromgaining a clear knowledge of it” [AT X, 395; transl. Descartes, 1985–1991, I, 29; on this example,see below Note 10]. Since Beeckman, explicitly answering a question from Descartes, found ademonstration that the hyperbola satisfies the condition and, after receiving Descartes’ approba-tion, transcribed it in a page of his Journal dated 1 February 1629, this would mean that thepassage could plausibly have been written between 1626 (when Descartes started his research onthis topic) and 1629.

4 The Progymnasta de solidorum elementis is a treatise on solid geometry and polyhedral numbersthat exists now only in a (presumably partial) copy with no clear indication of date. It wasreproduced in the Adam–Tannery edition [AT X, pp. 257–277]. Two more up-to-date editions areavailable, one with English translation [Descartes, 1982] and the other with French translation[Descartes, 1987].

5 Fol. 339v, dated 1 February 1629. At the end of the text, Beeckman writes: Hanc inventionem tantifacit D. des Chartes, ut fateatur se nihil unquam praestantius invenisse, imo a nemine unquampraestantius quid inventum [Beeckman, 1939–1953, Vol. IV, pp. 138–139].

6 See for instance [Israel, 1998]. To avoid anachronistic designations such as “analytical geometry”

or “algebraic geometry,” I shall use Descartes own terminology in La Géométrie: “Mais, parce quej’espère que dorénavant ceux qui auront l’adresse de se servir du calcul géométrique ici proposé, netrouveront pas assez de quoi s’arrêter touchant les problèmes plans ou solides, je crois qu’il est àpropos que je les invite à d’autres recherches, où ils ne manqueront jamais d’exercice” [AT VI, 390].


2. What Descartes said to Beeckman in October 1628

2.1. Situating the Cartesian program in 1628

According to Beeckman’s Journal, in 1628 “Louis René Descartes du Perron” went firstto Middleburg (the town where Descartes, in May 1619, had last sent letters to him) inorder to visit him. But Beeckman was no longer in Middleburg and it was not until8 October, 1628 that Descartes finally met his friend again in Dordrecht [Beeckman,1939–1953, Vol. III, p. 95, fol. 333r]. As he would reiterate nine years later,7 the Frenchmanexplained to his friend that he had worked a lot in arithmetic and geometry since their lastexchange and made such great advances in these sciences that he had nothing more toexpect from them (se in arithmeticis and geometricis nihil amplius optare).8 To make his pro-gress clear, he presented some “samples” of his work (cujus rei non obscura mihi speciminareddidit) and promised to send later a treatise of Algebra completed in Paris.

Even if Descartes did say that the treatise on algebra was complete, it was not yet readyfor publication because, as he explained, he thought of it as a basis for a more ambitiousproject intended to embrace all human knowledge.9 This ambitious project was, in fact,the main reason that Descartes gave for his visit. What he expected from Beeckman wasto engage in this program, pursuing their former fruitful collaboration: “traveling throughGermany, France and Italy, he had not found anybody else, he says, with whom he coulddiscuss according to his heart (secundum animi) and from whom he could hope for aid in hisresearches”.10 This development is transcribed by Beeckman in his journal under the title

7 In 1637 Descartes published the “Discours de la méthode pour bien conduire sa raison etchercher la vérité dans les sciences” as an introduction to a collection of “Essais de la Méthode”

presenting his major scientific achievements and consisting of La Dioptrique, Les Météores andLa Géométrie [AT VI].

8 Compare with Discours de la méthode, AT VI, 20–21. Just after recalling the results he hadobtained following the rules of his method, Descartes also explains in the Discours in what sense thiscould be seen as “perfect knowledge”: “In this I might perhaps appear to you to be very vain if youdid not remember that having but one truth to discover in respect to each matter, whoever succeedsin finding it knows in its regard as much as can be known” [Descartes, 2003, p. 94].

9 Paulo post Parisiis suam Algebram, quam perfectam dicit, quâque ad Geometriae scientiampervenit, imo qua ad omnem cognitionem humanam pervenire potest, propediem ad me missurus[Beeckman, 1939–1953, Vol. III, p. 95].10 Beeckman, 1939–1953, Vol. III, p. 95. This is not mere courtesy, as Descartes did indeed askBeeckman for help in the resolution of an important problem in optics: the proof of the fact that thehyperbola is an anaclastic curve. “The line called the ‘anaclastic’ in optics,” to repeat Descartes’ owndescription in the Regulae, is “the line from which parallel rays are so refracted that they intersect ata single point” [AT X, 393–394; transl. Descartes, 1985–1991, I, 28–29]. Immediately after the sampleof algebra, Beeckman also transcribed in his Journal other Cartesian works concerning optics andconic sections. The first fragment concerns the angle of refraction (Angulus refractionis Des Cartesexploratus fol. 333v) and contains the famous “law of sines.” The second concerns “burning lenses,”which Descartes discovered using the fact that the hyperbola is an anaclastic curve (Quod attinet adinventionem hyperbolicae sectionis ejus generis, per quam omnes radii in idem punctum refringantur,quod dictus DES CHARTES dicit se fecisse fol. 334r). This last fact was apparently not proved byDescartes, who asked Beeckman if he could demonstrate it. This is testified by a fragment dated1 February 1629, which begins with these words: “Hanc de hyperbola propositionem D. des Chartesindemonstratam reliquerat, ac me rogavit ut ejus demonstrationem quaererem, quam cum invenissem,gravisus est ac genuinam esse judicavit” [Beeckman, 1939–1953, Vol. III, pp. 109–110, fol. 338v].

432 D. Rabouin

Historia Des Cartes ejusque mecum necessitudo and is followed by a general commentary,entitled docti cur pauci, making reference to the fact that there were so few learned menat the time — a phrase which reminds us of the famous physico-mathematici paucissimiwritten by the same Beeckman when he first met Descartes in 1618 [Beeckman, 1939–1953, Vol. I, p. 244, fol. 100v].

The Algebra is hence presented in the context of a more ambitious project, which is veryclose to the one presented in the Regulae, in which Descartes warns,

11 Afalsitdata12 “Pfonde

I would not value these Rules so highly if they were good only for solving those pointlessproblems with which arithmeticians and geometers are inclined to while away their time,for in that case all I could credit myself with achieving would be to dabble in trifles withgreater subtlety than they. I shall have much to say below about figures and numbers, forno other disciplines can yield illustrations as evident and certain as these. But if oneattends closely to my meaning, one will readily see that ordinary mathematics is far frommy mind here, that it is quite another discipline I am expounding, and that these illustra-tions are more its outer garments than its inner parts. This discipline should contain theprimary rudiments of human reason and extend to the discovery of truths in any fieldwhatever. Frankly speaking, I am convinced that it is a more powerful instrument ofknowledge than any other with which human beings are endowed, as it is the sourceof all the rest. [Rule IV, AT X, 373–374; transl. Descartes, 1985–1991, I, 17]

The whole treatise, according to a general plan exposed in Rule XII, was supposed to con-tain a description of a general method to solve any question and was intended to consist ofthree parts of twelve rules each, dealing respectively with “simple propositions,” “perfectquestions,” and “imperfect questions.”11 In introducing the second part, Descartes gives awarning similar to that in Rule IV: “This part of our method was designed not just for the sakeof mathematical problems; our intention was, rather, that the mathematical problems shouldbe studied almost exclusively for the sake of the excellent practice which they give us in themethod.” [Rule XIV, AT X, 442; transl. Descartes, 1985–1991, I, 59: my emphasis.]

All these documents converge toward the idea that Descartes left France with a project thatwas supposed to go much further than mathematics, or even “physico-mathematics,” andtouch ad omnem cognitionem humanam. In the third part of Discours de la méthode (1637),he claims to have spent nine years after 1619 practicing “in the solution of mathematical prob-lems according to the Method, or the solution of other problems which though pertaining toother sciences, I was able to make almost similar to those of mathematics” [AT VI, 30]. ButDescartes then adds a very important comment: this occurred, he says, before he had taken“any definite part in regard to the difficulties as to which the learned are in the habit of dis-puting, or had commenced to seek the foundation of any philosophy more certain than thevulgar” [AT VI, 30]. As regard these “foundations,” the correspondence of 1629–1630 makesit clear that they were related to his later investigations in physics.

In a letter of October 1629 to Mersenne, Descartes states indeed, just after mentioninghis actual project of a small treatise of physics (on meteors), that he has “now taken a standon all the foundations of philosophy.”12 In a famous letter of 15 April 1630, he answers atheological question from Mersenne by stating,

“question” is everything in which a truth can be sought (as opposed to “intuition” in which noy can occur). A question is “perfect” when everything that is sought can be deduced from theand “imperfect” if not [AT X, 431].our la Raréfaction je suis d’accord avec ce Médecin et ai maintenant pris parti touchant tous lesments de la Philosophie” [AT I, 25: my emphasis].

13 “Mparticétabl14 “Pmaisprend15 In1630,“I halearnmodisee [G


I think that all those whom God has given the use of this reason have an obligation toemploy it principally in the endeavour to know him and to know themselves. That is thetask with which I began my studies; and I can say that I would not have been able to dis-cover the foundations of physics if I had not looked for them along that road. It is the topicwhich I have studied more than any other and in which, thank God, I have not altogetherwasted my time. At least I think that I have found how to prove metaphysical truths in a man-ner which is more evident than the proofs of geometry — in my opinion, that is: I do not knowif I shall be able to convince others of it. During my first nine months in this country I workedon nothing else. [AT I, 144; Descartes, 1985–1991, III, 22: my emphasis]

Hence it is clear that the first nine months that Descartes spent in Holland, in 1629, werededicated to these kinds of inquiries (foundations of physics, connected more generallyto the foundations of philosophy and the status of “metaphysical truths”) and that theyled to a decisive change in Descartes’ thought (“une prise de parti,” as he says). The veryfact that the evidence of geometry could be presented in 1629 not as a basis for a reformof all human knowledge as in 1628, but as inferior to “metaphysical truths,” is a strikingsign of this change. It is in accordance with the famous thesis presented in the correspon-dence of 1630 with Mersenne concerning the creation of eternal (i.e., mathematical) truthsby God.13 In the letter of 6 May, Descartes emphasizes,

Those who have no higher thoughts than these can easily become atheists; and becausethey perfectly comprehend mathematical truths and do not perfectly comprehend thetruth of God’s existence, it is no wonder they do not think the former depend on the latter.But they should rather take the opposite view, that since God is a cause whose powersurpasses the bounds of human understanding, and since the necessity of these truthsdoes not exceed our knowledge, these truths are therefore something less than, and sub-ject to, the incomprehensible power of God. [To Mersenne, AT I, 150; transl. Descartes,1985–1991, III, 25: my emphasis]

Moreover, Descartes explains that he is no longer interested in mathematics14 and mentionssome treatises begun in Paris that he left unfinished because he realized that they weredependent on other important issues that he had to deal with first [AT I, 138].

One should be careful not to conflate these different steps: a first project, presented toBeeckman at the end of 1628, was already supposed to lead to a more general programconcerning “all human knowledge,” but it did not imply research on the “foundations ofphilosophy”; when Descartes finally took a stand on these foundational issues, at the begin-ning of 1629, he also clearly changed his mind considering the role of mathematical certaintyand the feasibility of a pure Mathematica-Physica.15

ais je ne laisserai pas de toucher en ma Physique plusieurs questions métaphysiques, etulièrement celle-ci: Que les vérités mathématiques, lesquelles vous nommez éternelles, ont été

ies de Dieu et en dépendent entièrement, aussi bien que tout le reste des créatures” [AT I, 145].our des problèmes, je vous en enverrai un million pour proposer aux autres, si vous le désirez;je suis si las des mathématiques, et en fais maintenant si peu d’état, que je ne saurais plusre la peine de les soudre moi-même” [AT I, 139].this regard, one should remember that on the occasion of his brief rupture with Beeckman, inDescartes criticized very violently what his friend had achieved under this particular program:

ve never learnt anything but idle fancies from your Mathematico-Physica, any more than I havet anything from the Batrachomyomachia” [AT I, 159; Descartes, 1985–1991, I, 27, translationfied]. On the complex relationship between Descartes and “Physico-mathematics” after 1630,arber, 2000].

434 D. Rabouin

2.2. Ars generalis, scientia penitus nova, Mathesis universalis, etc

In the above I have drawn attention to the complexity and the richness of Descartes’projects during the years 1628–1629. As we have seen, documents exist, be they from thisvery period (Beeckman’s Journal, correspondence of 1629–1630) or from later memories(Discours de la méthode, 1637), testifying that Descartes changed his mind in a significantway during this time. I now introduce another document often mentioned in the literaturebut to my mind less directly connected with our topic: the letter to Beeckman of 26thMarch 1619. In it, Descartes announced to his friend, not long before leaving Holland, thathe wanted to launch a “completely new science (scientia penitus nova) by which all questionsin general may be solved that can be proposed about any kind of quantity, continuous aswell as discrete. But each according to its own nature.” He then makes a parallel:

16 Desolvetrulythis showequaes

In arithmetic, for instance, some questions can be solved by rational numbers, some bysurd numbers, and others can be imagined but not solved. For continuous quantity Ihope to prove that, similarly, certain problems can be solved by using only straight orcircular lines, that some problems require other curves for their solution, but still curveswhich arise from one single motion and which therefore can be traced by the new com-passes, which I consider to be no less certain and geometrical than the usual compassesby which circles are traced; and, finally, that other problems can be solved by curved linesgenerated by separate motions not subordinate to one another. [AT X, 157–158; transl.Bos, 2001, p. 232]16

This program, still in infancy and merely analogical, is nonetheless very close to that ofLa Géométrie, in which Descartes succeeded in proposing a new classification of geometricalproblems through a classification of curves (relying on the one hand on what he calls“continuous motion” and on the other hand on algebraic criteria). It is therefore very tempt-ing to consider that it was the core of “the” Cartesian program from the encounter withBeeckman in 1618 to the Discours de la méthode and the Essais appended to it in 1637.The “general algebra” of 1628 could then be considered as a step on this continuous path.

But this kind of continuity seems to me a mere artefact coming from the tendency to gofrom the programs to a reconstruction of the practice and not from the actual practice to anevaluation of the programs. I shall indicate this more clearly in the following pages. Buteven staying at the level of declarations, one should emphasize the fact that neither inthe documents given by Beeckman in 1628–1629, nor in the Regulae, is there any mentionof a classification of geometrical problems depending on the classification of curves. Howshould we interpret this silence if this is supposed to be at the very core of Descartes’ pro-gram? The study of curves (in fact, only conic sections) is presented in 1628 separately andindependent of the “general algebra” in the context of works in optics. Moreover, Descartesdoes not use algebra in his geometrical analysis of these curves, as transcribed byBeeckman.

One should also be very cautious when identifying the program of the new discipline(alia disciplina) announced in the Regulae, in the first part of Rule IV, with the mathesis

scartes then continues, “But in due time I hope to prove which questions can or cannot bed in these several ways: so that hardly anything would remain to be found in geometry. This is ainfinite task, not for a single person. Incredibly ambitious; but through the dark confusion ofcience I have seen some kind of light, and I believe that by its help I can dispel darknessver dense.” In the margin, Beeckman writes ars generalis ad omnes quaestiones solvendaita.


universalis mentioned in the second part of the same Rule IV, to ground a hypotheticalproximity with the program of a scientia penitus nova of 1619 (which was supposed to“solve any question that can be proposed about any kind of quantity, continuous as wellas discrete”). Without entering into the details of this controversial matter,17 I simply recallwhat Descartes says in Rule IV. The general context is an investigation into what makespossible the unity of mathematics — which is indeed the traditional context of reflectionon a possible “general” or “universal” mathematics. On this path, Descartes arrives atthe observation that “the exclusive concern of mathematics is with questions of order ormeasure and that it is irrelevant whether the measure in question involves numbers, shapes,stars, sounds, or any other object whatever.” This leads him to the conclusion “that theremust be a general science (scientia generalis) which explains all the points that can be raisedconcerning order and measure irrespective of the subject-matter, and that this scienceshould be termed not with an unusual name, but with a venerable one, with a well-estab-lished meaning: mathesis universalis — for it covers everything that entitles these other sci-ences to be called branches of mathematics” [AT 377, 9–378, 10; Descartes, 1985–1991, I,19]. The mathesis universalis is therefore presented as a “venerable” discipline with a“well-established meaning,” not a “completely new science.” A few lines later, Descartesgoes further by stating that everybody not only knows its name, but also its subject matter(cum nomen ejus omnes norint, et, circa quid versetur... intelligant) and he then wonders whypeople do not pay more attention to it [Descartes, 1985–1991, I, 20]. His answer marks thesecond (and last!) use of the term in the entire Cartesian corpus:

17 Fonotio18 Domihi

Aware how slender my powers are, I have resolved in my search for knowledge of thingsto adhere unswervingly to a definite order, always starting with the simplest and easiestthings and never going beyond them till there seems to be nothing further which is worthachieving where they are concerned.18 Up to now, therefore, I have devoted all my ener-gies to this mathesis universalis, so that I think I shall be able in due course to tackle thesomewhat more advanced sciences, without my efforts being premature. [AT X, 378–379,6; Descartes, 1985–1991, I, 20; transl. modified]

Mathesis universalis is therefore a discipline that Descartes studied before he dared to“tackle the somewhat more advanced sciences,” according to a rule that states that oneshould never go beyond the easiest matter “till there seems to be nothing furtherwhich is worth achieving where they are concerned” — the exact contrary of a programindeed.

My aim here is to draw attention to the fact that there is no reason to merge what seemto be different projects (the “entirely new science” of 1619, the alia disciplina and themathesis universalis of the Regulae, the “general algebra” of 1628) into a single grandioseview culminating in La Géométrie. In fact, it seems very dubious to ground any hermeneuticpostulate in these texts at all, since they are very allusive and surrounded, as we saw in theprevious section, by many other texts calling into question the continuity of Descartes’ lineof thought. In my opinion, the only way to solve the uncertainties raised by these texts is toconfront them with Descartes’ actual mathematical practice.

r a more detailed analysis, see [Rabouin, 2009]. For an overview of the debates around thisn, see [Van de Pitte, 1991] and [Sasaki, 2003, Chap. IV § 3].nec in istis nihil mihi ulterius optandum superesse videatur: note the proximity with the Is dicebat

se in arithmeticis and geometricis nihil amplius optare.

436 D. Rabouin

2.3. The general algebra

I turn now to the Algebrae Des Cartes specimen quoddam (henceforth the Specimen), thetext transcribed by Beeckman, which describes a “general algebra” invented by Descartes(dicit idem se invenisse Algebram generalem). It relies on a simple geometrical schematismin which Descartes uses only plane and not solid figures (se non uti corporum figuris, sedplanis duntaxat). This echoes a famous passage of the Regulae in which Descartes explainshow he distanced himself from the traditional interpretation of algebraic powers corre-sponding to geometrical dimensions, because he understood that they were “nothing butmagnitudes in continued proportion” and, as a consequence, “should never be representedin the imagination otherwise than as a line or a surface.”19

But, as we shall see, this is not the justification of the notion of dimension given in ourtext. Moreover, the justification in the Regulae seems closely tied to what Descartes calledthe “number of relations,” which is usually interpreted as connected with the use of anexponential notation. However, another surprising element of our text is that it still usescossic notation.20 The point is not that Descartes uses this notation in and of itself, but thathe does so when introducing what is supposed to be the results of his great advances inmathematics during the nine preceding years — a very intriguing situation if he was in pos-session of his new notation at that time. One could of course attribute this “archaism” tothe copyist and make the hypothesis that Beeckman changed Descartes’ notation, but thiswould mean, in any case, that Descartes did not emphasize its importance. On the otherhand, one should be very cautious when dealing with Descartes’ notation in the Regulaeand not forget that we possess only late copies of the text, made after the publication ofLa Géométrie. Besides, these copies are not coherent with regard to notation. For example,Rule XVIII does not satisfy the prescription presented in Rule XVI to represent indetermi-nate quantity by lower case and unknown by uppercase, and uses indifferently a and A todesignate the same quantity. One striking example of the modifications introduced by edi-tors is in the Excerpta mathematica, a collection of mathematical essays by Descartes thatwas published for the first time in the same volume as the Regulae: Renati Descartes Opus-cula posthuma [Descartes, 1701]. We possess indeed other copies of some texts than the onegiven in the Opuscula and one can clearly see in them that x and xx were substituted for

19 “We should note also that those proportions which form a continuing sequence are to beunderstood in terms of a number of relations; others endeavour to express these proportions inordinary algebraic terms by means of many dimensions and figures (. . .). I realized that suchterminology was a source of conceptual confusion and ought to be abandoned completely. For agiven magnitude, even though it is called a cube or the square of the square, should never berepresented in the imagination otherwise than as a line or a surface. So we must note above all thatthe root, the square, the cube, etc. are nothing but magnitudes in continued proportion which, it isalways supposed, are preceded by the arbitrary unit mentioned above” [AT X, 456–457; Descartes,1985–1991, I, 68].20 “Cossic” comes from Cristopher Rudolff’s book Coss (1525) — which means “unknown” (likethe Italian cosa), the “cossic art” being then the art of dealing with unknowns. This style of algebraspread in Germany in the 16th century (with authors like Stifel, Faulhaber, or Roth, to whom I shallcome back later). It relied on a notational system in which a symbol was used for every power of theunknown. The notation transcribed by Beeckman is for the root, for the square, for the cubeand for the square of the square (biquadratum).

Fig. 1. Facsimile of fol. 333 of Beeckman’s autograph. Excerpted from (Beeckman, 1939–1953, III).


what, in other sources, are cossic symbols.21 One could also note that the Opuscula uses acossic sign for addition z in Rule XVI, which was changed in the Adam–Tannery edition. Ishall come back to these issues later because the first surprise occurs before one evenreaches them: in the presentation of the schematism.

As observed above, the schematism used by Descartes in the “general algebra” relied ontwo-dimensional figures — the reason invoked being that this representation is easier tograsp (qui eae facilius mentibus insinuantur). According to Descartes, one should representthe unit by a small square, the root by a parallelogram composed of as many times thesquare unit as needed, and then the square, the cube or the square of the square as rectan-gles composed of as many times the roots as implied in them. In the diagram drawn byBeeckman (Fig. 1), a is the square unit, b the root composed of three units, c the squarecomposed of three times three units, d the cube composed of three times the square[Beeckman, 1939–1953, Vol. III, p. 95, fol. 333r].

Beeckman explains that the representation could be done with simple lines only (pernudas lineas), each representation being prefixed by the cossic symbol corresponding to it(Fig. 2).

This could be seen as an anticipation of the famous and groundbreaking opening ofLa Géométrie, in which Descartes explained that all the operations of arithmetic couldbe interpreted as manipulations on “simple lines.”22 However, certain details of the text pre-

21 See [AT X, 298]. Descartes’ discourse about the “number of relations” in the Regulae is perfectlycompatible with other notation, such as Stevin’s (which Descartes could have come across in [VanRoomen, 1597] — a plausible source for his concept of mathesis universalis; see [Crapulli, 1969]) oreven an adaptation of [Clavius, 1609]. It should indeed be noted that Clavius (like German cossists:Stifel, Roth, Faulhaber) proposed, when introducing several unknowns, to mark them by capitalletters A, B, C — in the same way as Descartes proposes in Rule XVI [Clavius, 1609, Chap. XV,p. 72 sq]. Moreover, what we now write as “2y4” would be written by Clavius as “2A”, expressingthe power of an unknown and not an unknown in and of itself (p. 73). Since was referred to, at thevery beginning of Clavius’ treatise, as corresponding to the “exponent” 4, it would have beenvery natural (and convenient) to abridge this first notation as “2A4”, or “2A(4)” (Girard’snotation) if one wanted to avoid ambiguity (since in Clavius’s notation “2A” could refer to “2y4”

or “2yx4”).22 [AT VI, 371]: “il est à remarquer que, par a2 ou b3 ou semblables, je ne conc�ois ordinairement quedes lignes toutes simples, encore que, pour me servir des noms usités en l’algèbre, je les nomme descarrés, ou des cubes, etc.” On the role of this “calcul des segments” in Descartes’ Géométrie, see[Jullien, 1996].

Fig. 2. Representation per nudas lineas. Facsimile of fol. 333 of Beeckman’s autograph excerptedfrom (Beeckman, 1939–1953, III).

438 D. Rabouin

vent us from jumping to this conclusion. When introducing the representation through unitsquares, Descartes did indeed add a strange ita etiam punctum concipit (“or he conceives ofit in the same way with a point”). After completing the first sequence, Beeckman statesmore explicitly: “Or rather, he explains also all of this by lines so that a would representa point, b a line, c a square and d a cube. In this manner f represented a cube producedby the multiplication of the square e by the number of roots.” [Beeckman, 1939–1953,Vol. III, p. 95, fol. 333r: my emphasis.]

This whole development is very close to Stevin’s ideas. In his Arithmétique, first publishedin 1585, and reedited by Albert Girard in 1625, Stevin insisted on the fact that one couldinterpret algebraic powers in terms of geometric magnitudes in a continuous proportionand therefore proposed a very simple geometric schematism, in which there would be noneed to escape from the three dimensions of everyday space. The main difference betweenDescartes and Stevin is that Stevin then used a three-dimensional schematism (made of“beams”, which he calls “plinthides” and “docides”, and not of rectangles).23 See Fig. 3.

But the fact that Descartes also allows us to represent the unit by a geometrical point isastonishing. It goes directly against what Stevin rightly designated as the great mistake ofhis predecessors, who considered the unit in arithmetic as somehow equivalent to the pointin geometry [Stevin, 1955–1966, Vol. II B, p. 498].

The parallel with Stevin leads to another important remark: as in Stevin, the use of theschematism is here not operative, but foundational,24 that is, there is nothing like a geomet-rical calculus in our text, and Descartes does not explain how to construct the differentmagnitudes through determinate operations. This is not merely an argument from silencesince there seems to be, at first sight, no possibility of grounding a geometric algebra ifthe unit is represented as a point and the root as a line.

The apparent confusion of Descartes is reinforced by the following passage, in which heexplains how one should understand the notion of algebraic dimension:

23 Thdunta24 “DStevi

He conceives the cube in particular through three dimensions, as do others. But he con-ceives a biquadratum as if a simple cube made of wood was transformed into a stonecube: in this manner, indeed, one dimension is added to the whole; but if another dimen-sion is to be added, he considers a cube made of iron; then of gold, etc. — that which canbe obtained not only with weight but also with colours and with all the other qualities.Cutting a wooden cube in three squares, he thus conceives that he is cutting a cube madeonly of wood, of stone etc. so that the iron cube is transformed into the wooden cube in

is could provide a way of interpreting the opening se non uti corporum figures, sed planisxat.escription du fondement des nombres géométriques” is in fact the title of the section in which

n introduces his schematism [Stevin, 1625].

Fig. 3. Representation of the powers of 2 according to Stevin [from pp. 12 and 13 of hisArithmetique 1585; reproduced in Stevin, 1955–1966, Vol. IIB, pp. 511–512].

25 Ontôn c1609]naturinterp


the same manner than the simple cube is into the squares which are to be considered ineach kind. [Beeckman, 1939–1953, Vol. III, p. 96, fol. 333v, my transl.]

One could consider that these developments provide just one among many other examplesin the history of science of a powerful technique being attached to confused conceptualgrounds. Unfortunately, we have here no evidence of a powerful technique. Indeed, in the fol-lowing passage, Descartes shows his friend evidence of his technique. But what example doeshe choose? The resolution of a quadratic equation through a simple manipulation of squares,that is to say one of the most elementary techniques in algebra, known since Al-Khwarizmi!

As one can see in Fig. 4, to solve the equation x2 = 6x + 7, Descartes constructs thesquare of side x, divides the coefficient by 2 to remove two rectangles of area 3x, and thusobtains a new square DE. In this construction, the square of side GD is subtracted twiceand one has to compensate for this by adding 9 to the whole. So if one subtracts 6x fromthe sought square x2, one will obtain 7 + 9 = 16. This gives us the area of square DE (16),and then its side DF = 4. Since DG = 3, one obtains the result AC = 7. This is the (only)“sample” of Descartes’ technique given in 1628! And it is presented as the result of nineyears of mathematical studies in which “he has made as much progress as was possiblefor a human mind”!

Then Beeckman notes, “The irrational numbers, which cannot be explained otherwise, heexplains them by a parabola (per parabolam).” This allusive statement could be a referenceto the solution of the third- and fourth-degree equations by the intersection of a parabolaand a circle, which Descartes sent to his friend four months later and which he may haveannounced to him in October.25 The end of the text contains a discussion of the different

e should, however, keep in mind that the Greek name for “application of area” was parabolêhoriôn. This technique, exposed in Book II of Euclid’s Elements, was presented by [Clavius,when dealing with the extraction of square roots (Chap. XII, p. 46 sq.). As we will see, it was

ally involved in the calculus presented in the Regulae, so one should not dismiss thisretation of per parabolam too quickly.

Fig. 4. The resolution of an equation in 1628. (Beeckman, 1939–1953, III p. 96).

440 D. Rabouin

denominations of the roots of an equation (“true”, “implicit” (i.e., negative), and “imagi-nary”) and the number of them, which Descartes derives, he says, ex tabula vulgari(fol. 333v).

By recalling the Algebrae Des Cartes specimen quoddam I want to emphasize the manyweaknesses it presents: the geometric schematism does not seem well grounded, the alge-braic technique is trivial, and the notion of algebraic dimension is metaphorically explainedthrough a qualitative analogy. How are we to understand this collection of disappointingfeatures if Descartes at that time was already in possession of an original and powerfultechnique? Was this the outcome of an “entirely new science”?26 Not that a great mathe-matician cannot make errors, feel hesitations, or have zones of confusion in his conceptualfoundations. But we have to keep in mind that we are not dealing here with a draft thatDescartes kept for himself. What we have here is nothing less than a demonstration broughtto make clear the progress made during the nine preceding years and to convince Beeckmanto engage in a more ambitious program.

Of course, one could object that the poverty and the confusion that I have attributed toDescartes were in fact linked to the very specific situation in which this text arose. MaybeDescartes was simply hiding from Beeckman the real power of his technique; maybe headapted a powerful technique to a simple example, taking into account the low mathemat-ical level of his interlocutor; maybe Beeckman, who was certainly not an expert algebraist,simply did not understand what Descartes had explained to him; maybe he introduced hisown confusion into Descartes’ clear and distinct ideas, etc.

These arguments are very difficult to dismiss since they rely on the postulate that there issomething behind the extant documents, about which we have no direct evidence at all. Thisis why I presented my interpretation as a methodological exercise. There are nonethelesspositive arguments that indicate that Beeckman’s mathematical weaknesses are insufficientto justify the disappointing features presented by the sources. It should be noted first thatDescartes is truly asking Beeckman for help in the solution of a “physico-mathematical

26 See for example [Sasaki, 2003, p. 166]: “What can be asserted from our examination ofBeeckman’s record ‘A certain specimen of Descartes’s Algebra’ is that before October 1628,Descartes had already possessed his own system of algebra, which can be thought to be the producton an ‘entirely new science.’”


problem” and that there is no reason to suspect that he is insincere when saying that hewants to launch a new ambitious program with him or that he is hiding something fromhim. Suppose, for the sake of the argument, that Descartes did not show the full extentof his skills to his friend. It would not change the essence of our interrogations: why didhe not even mention what is supposed to be the real content of his techniques (the exponen-tial notation, the classification of curves, the algebraic analysis of geometrical problems)?Remember that Descartes had no difficulty in announcing his ambitious program of a “newscience” in 1619. It would be very natural to mention the outcome of this program, if therewas any. Besides, why would Descartes think that the treatment of quadratic equations is asimple example of his techniques? Finally, it should be pointed out that Descartes had nodifficulty sending a much more complicated example (the resolution of solid problemsthrough the intersection of a circle and a parabola) to his friend a few months later; onceagain, as we shall see below, we find no trace of the exponential algebraic notation in thispiece, nor of the classification of curves, nor of the algebraic analysis of geometrical prob-lems. In this case, there is no reason to suspect that Beeckman made changes to Descartes’writings, since he explicitly states that he made a faithful copy of them (quod ex illius scriptisad verbum describo). Last, but not least, we possess an independent source supporting ourinterpretation: the same confusions on similar topics appear in the Regulae ad directionemingenii.

3. Dimensio, unitas, et figura: looking outside of the Specimen

3.1. The schematism of the Regulae

There is extensive literature on Descartes’ Rules for the Direction of the Mind, but stran-gely, some passages seem to persistently escape the attention of readers. These passages pre-cisely concern the three notions identified above: dimension, unity, and figure (dimensio,unitas et figura), and the fact that two schematisms are proposed by Descartes — dependingon which representation one chooses for the unit. Therefore, I would like to put particularemphasis on these aspects. In Rule XIV Descartes famously introduces the foundation ofhis geometrical calculus (presented in Rule XVIII) by stating that “in all reasoning it is onlyby means of comparison that we attain an exact knowledge of the truth” and that “the busi-ness of human reason consists almost entirely in preparing this operation.” This establishesthe central role of proportion in the treatment of any “question” and the main goal of themethod in this second part of the treatise: to reduce these proportions to simple compari-sons, that is, equalities [AT X, 440; Descartes, 1985–1991, I, 57–58]. According to this gen-eral structure, any “question” can be expressed in terms of proportions between magnitudesand, in consequence, through some spatial representation “since no other subject [scil. thanspatial extension] displays more distinctly all the various differences in proportions.” Hencethe conclusion that “perfectly determinate problems present hardly any difficulty at all,save that of expressing proportions in the form of equalities, and also that everything inwhich we encounter just this difficulty can easily be, and ought to be, separated from everyother subject and then expressed in terms of extension and figures. Accordingly, we shalldismiss everything else from our thoughts and deal exclusively with these until we reachRule twenty-five” [AT X, 441; Descartes, 1985–1991, I, 58].

These ideas are taken up again by Descartes in the Discours de la méthode, when hecomments on the very first application of the Method. He starts by recalling, as in RuleIV, that he studied the common part of mathematics, “observing that, although their

Fig. 5. How to represent the difference between white, blue, and red according to Rule XII [fromDescartes, 1701, p. 34].

442 D. Rabouin

objects are different, they do not fail to agree in this, that they take nothing under consid-eration but the various relationships or proportions which are present in these objects, Ithought that it would be better if I only examined these proportions in their general aspect.”He then explains the way in which he could handle these proportions:

27 [A28 On

Then, having carefully noted that in order to comprehend the proportions I should some-times require to consider each one in particular, and sometimes merely keep them inmind, or take them in groups, I thought that, in order the better to consider them indetail, I should picture them in the form of lines, because I could find no method moresimple nor more capable of being distinctly represented to my imagination and senses.I considered, however, that in order to keep them in my memory or to embrace severalat once, it would be essential that I should explain them by means of certain formulas,the shorter the better. And for this purpose it was requisite that I should borrow all thatis best in Geometrical Analysis and Algebra, and correct the errors of the one by theother.27

The main difference between the two documents is that, in the second case, the “method”is presented in the context of an application to pure mathematics or physico-mathematica,whereas the Regulae are concerned more with epistemological considerations (with explicitreference to the theory of imagination exposed in Rule XII). In particular, one shouldnotice that the model presented in Rule XIV on the example of colors (see Fig. 5) is notyet that of a geometrical calculus stricto sensu, but that of a schematic representation usedto code differences.28 This is clear from the beginning of Rule XII, where Descartes treatsthe example of colors, which he alludes to in Rule XIV.

As I shall describe in the following paragraphs, Descartes explains in detail the epistemo-logical grounds of his geometric schematism, expanding at large on the definition of “exten-sion.” He then arrives at “the characteristics of extension itself which may assist us inelucidating differences in proportion”, namely our three key notions: dimension, unity,and figure.

Dimension, first, is defined in term of the number of parameters entering a question.Hence a triangle can be seen as having three dimensions, since it needs three parametersto fully determine it (the three sides, or two sides and an angle, or two angles and its area).

T VI, 19–20; Descartes, 2003, p. 93].this general project, see [Sepper, 1996, Chap. III] and [Fichant, 1998, pp. 1–28].


In the same way, a trapezium can be seen as having five dimensions; a tetrahedron, six, etc.The difference from the “qualitative” interpretation presented in the Specimen is that herethe notion is more clearly limited to measurable parameters: “By ‘dimension’ we mean sim-ply a mode or aspect in respect of which some subject is considered to be measurable. Thuslength, breadth, and depth are not the only dimensions of a body: weight too is a dimension— the dimension in terms of which objects are weighed. Speed is a dimension — the dimen-sion of motion; and there are countless other instances of this sort” [AT X, 447; Descartes,1985–1991, I, 62]. This is consistent with the idea that Descartes’ overall project was at thattime still close to Beeckman’s notion of a Physico-mathematica, and there is no clear dis-tinction between the mathematical and the physical notions of dimension (what he saysbeing close to what we now deal with in “dimensional analysis”).

Of particular interest is the fact that Descartes then justifies the limitation to two dimen-sions not through mathematical requisites but according to his theory of knowledge [AT X,449; Descartes, 1985–1991, I, 63]. But the most interesting development comes with thenotion of unity considered as a “common nature which all the things which we are compar-ing must participate in equally.” Descartes goes on to say that “we may adopt as unit eitherone of the magnitudes already given or any other magnitude, and this will be the commonmeasure of all the others.” And he then adds a significant comment: “We shall regard it ashaving as many dimensions as the extreme terms which are to be compared. We shall con-ceive of it either simply as something extended, abstracting it from everything else — inwhich case it will be the same as a geometrical point (the movement of which makes upa line, according to the geometers), or as some sort of line, or as a square” [AT X, 449–450; Descartes, 1985–1991, I, 63–64]. Notwithstanding the fact that Descartes states herethat the unit should have “as many dimensions as the extreme terms which are to be com-pared” — an intriguing statement to which I shall return — he seems to repeat here againthe “unfortunate” error pointed out by Stevin: a unit can be conceived either as a geomet-rical point or as a line segment or as a square. Furthermore, Descartes then justifies the dou-ble regime of representation in the following way:

As for figures, we have already shown how ideas of all things can be formed by means ofthese alone. We have still to point out in this context that, of the innumerable differentspecies of figure, we are to use here only those which most readily express all the variousrelations or proportions. There are but two kinds of things which are compared witheach other: multiplicity and magnitudes. We also have two kinds of figures which wemay use to represent these conceptually: for example, the points,

......

which represent a triangular number. . . . Figures such as these represent multiplicities;while those which are continuous and unbroken, such as4, h etc., illustrate magnitudes.[AT X, 450; Descartes, 1985–1991, I, 64; transl. modified]

At the beginning of the next Rule (XV), Descartes repeats, “If we wish to form more dis-tinct images of these figures in our imagination with the aid of a visual display, then it isself-evident how they should be drawn. For example, we shall depict the unit in three ways,viz. by means of a square, h, if we think of it only as having length and breadth; by a line,

, if we regard it as having just length; or, lastly, by a point, ., if we view it as the elementwhich goes to make up a multiplicity” [AT X, 453; Descartes, 1985–1991, I, 65–66, transl.modified].

Fig. 6. The geometrical calculus of the Regulae (Rule XVIII), as presented in the Opuscula Posthuma[Descartes, 1701, p. 64].

444 D. Rabouin

In the Regulae, however, the schematism is not confined to a foundational role, but leadsto an operative use presented in Rule XVIII.

As one can see in Fig. 6, the calculus is now presented only with segments and rectangles.This is certainly why many commentators did not pay much attention to the double regimeof representation presented in the preceding rules. One should also notice that this schema-tism is quite different from the one presented to Beeckman. In the Specimen, Descartes pre-sented what we now write as x3 as x times a “square,” a “square” being itself x times arectangle composed of x “unit squares.” He saw that this procedure could be accomplishedonly per nudas lineas so that the square would be x times a line composed of x unit seg-ments. This procedure can be immediately applied to the product of any given magnitude,so that abc is c times b times a line composed of a unit segments. But this is not the pathfollowed in Rule XVIII. There, Descartes is much closer to the schema in use since the timeof the ancient Greeks (at least), in which one represents multiplication and division as form-ing rectangles and taking their side.29 This discrepancy supports well the distinctionbetween a foundational and an operative use of the schematism. Descartes used a certainschematism to figurate the quantities involved in a question (XIV–XV), but when it cameto computations proper, he went back to the representation of multiplication as forming

29 The main difference being, of course, that Descartes is not constrained by the problem ofdimensionality and can express the product of three magnitudes abc by taking a segment equal tothe rectangle ab and use it to form a rectangle with other side c.


rectangles (XVIII). This was, in fact, also the case in the Specimen when he solved thequadratic equation and represented 3x as a rectangle with sides x and 3.

The preceding passages should suffice to make it clear that the weaknesses of theSpecimen were due neither to some misunderstanding or lapsus calami coming fromBeeckman, nor to an oversimplification coming from Descartes. When Descartes waswriting the Regulae, he fell into the same apparent confusions concerning the notion ofdimension, the status of the unit and the way to represent it. This then raises two importantquestions, which I shall tackle in the remaining part of the paper: why did Descartesmaintain a double regime of representation, which looks at first like a useless source ofconfusion? How are we to understand the compatibility between what I have called the“foundational” and the “operative” use of the schematism?

3.2. Points and numbers

To illustrate the fact that one can use dots to represent “multiplicities”, Descartes givesthe example of “triangular numbers” and of a family tree [AT X, 450]. These examples arevery important if one wants to understand why Descartes could have been attached to thiskind of representation. If we cease to read the development of the years 1628–1629 throughthe lens of his future mathematics and ask ourselves, “what kind of mathematics did Des-cartes do before?” we see immediately that Descartes actually spent a lot of time and energyreflecting on “figurate numbers” and on polyhedra. In fact, the only extant Cartesian mathe-matical treatise from this period (Progymnasmata de solidorum elementis [AT X, 257–277],which we possess in a partial copy made by Leibniz) deals precisely with these questions, asdo some important passages of the Cogitationes Privatae.30 This thus allows a simpleanswer to our first question: Descartes maintained a double regime of representation simplybecause it corresponded to the kind of mathematics he was doing.

In agreement with P. Costabel [Descartes, 1987], I consider the Progymnasmata to havebeen conceived at the beginning of the 1620s. A number of arguments have been advancedto support this hypothesis, although none of them are completely conclusive. The use of cossicnotation seems to indicate an early writing (but there is an important exception in a text datedfrom 1638 [AT X, 297–299]). One can also add the fact that Descartes professed to Mersennein 1638 that he had neglected geometry for “more than fifteen years” [AT II, 99]. Since the firstpart of the copy made by Leibniz deals precisely with geometry, this claim would make theDe Solidorum elementis the outcome of studies made before 1623.31 Finally, there has recently

30 See [AT X, 241]: on triangular numbers; [AT X, 246–247]: on the rectangular tetrahedron; [AT X,247–248]: on the pyramid, and [Sasaki, 2003, pp. 128–132], for an English translation and adescription of these fragments.31 Descartes, 1987, pp. 106–109. This claim by Descartes does not preclude the fact that the treatisecould have been written later. Neither does it limit the kind of geometry that Descartes was doing inthe 1620s. In the letter to Mersenne, dated 15 April 1630, where Descartes confesses that he is nowtired of mathematics, he gives examples of other kinds of (plane) geometrical problems, which hesolved using only ruler and compass: “Invenire diametrum sph�r� tangentis alias quatuormagnitudine et positione datas. Invenire axem parabol� tangentis tres lineas rectas positione dataset indefinitas, cujus etiam axis secet ad angulos rectos aliam rectam etiam positione datam etindefinitam. Invenire stilum horologii in data mundi parte describendi, ita ut umbr� extremitas, data dieanni, transeat per tria data puncta, saltem quando istud fieri potest” [AT I, 139]. It should be notedthat these problems are all plane problems and that it is interesting that Descartes brought up thiskind of example if he was already dealing with much complex ones.

446 D. Rabouin

been a renewal of interest in the striking proximity between Descartes’ preoccupations inDe Solidorum elementis and research by contemporary German mathematicians such asJohannes Faulhaber, Johannes Remmelin, and Peter Roth.32 Even if one remains cautiousregarding the story (told by Lipstorp) of the encounter between Descartes and Faulhaberin Ulm in 1620, there are, as Schneider has summarized, “a number of astounding concur-rences in the treatment of various mathematical problems besides the solution of the generalquartic. These concern, for instance, the rule of signs for a determination of how many realroots an equation can have, a Pythagorean theorem for three dimensions, Faulhaber’sextension of figurate numbers to polyhedral numbers of the five regular polyhedra and thecontent of Descartes’ De solidorum elementis, which contains a relation between the numbersof corners, edges, and faces of regular polyhedra” [Schneider, 2008, p. 53].

To this set of converging arguments, I shall add another one. In the Discours, as we haveseen, Descartes presents the first outcome of his method as a study of mathematics, morespecifically of its unifying part (“ratios and proportions”). Then he adds,

32 Se2006]

As a matter of fact, I can venture to say that the exact observation of the few preceptswhich I had chosen gave me so much facility in sifting out all the questions embraced inthese two sciences [scil. Algebra and geometrical analysis], that in the two or three monthswhich I employed in examining them — commencing with the most simple and general,and making each truth that I discovered a rule for helping me to find others — not onlydid I arrive at the solution of many questions which I had hitherto regarded as most dif-ficult, but, towards the end, it seemed to me that I was able to determine in the case ofthose of which I was still ignorant, by what means, and in how far, it was possible tosolve them. [AT VI, 21; Descartes, 2003, p. 93]

According to this story, Descartes obtained the first results of his methodological inves-tigations in algebra and geometry as early as 1620. However, one should be cautious whenevaluating this declaration, since Descartes wrote it more than 15 years after the event. Inany case, we would have to explain how it is that Descartes was convinced in 1628 that“insofar as arithmetic and geometry were concerned, he had nothing more to discover.”In what kind of “Geometry” and “Arithmetic” did Descartes obtain his results? As I havetried to argue in the preceding section, the many weaknesses of the contents of the Specimenand of the Regulae should convince us that there is not necessarily a “hidden” technique touncover under this declaration. My hypothesis is that it most likely refers to works such asthe Progymnasmata, which present the advantage of being compatible with the weaknessesremaining in the texts of 1628. The De solidorum elementis would indeed fit perfectly intoDescartes’ description in the Discours, not only because they contain some new and inter-esting results in geometry (including an algebraic demonstration of the fact that there areonly five regular polyhedra — which was considered by many people at that time as theculmination of Euclid’s Elements), but also because the study of “figurate numbers” wasan important part of algebra in the German tradition.

3.3. Calculating with points

I will limit my study of the Progymnasmata to aspects of particular interest to us. Thecopy made by Leibniz does not contain any diagrammatic representation of figurate num-

e the studies by [Schneider, 1993 and 2008; Mehl, 2001; Penchèvre, 2004; Manders, 1995 and.


bers through dots, but this representation is implied by their definition and by the very aimof the second part of the text: to calculate the “weights” (i.e., the number of vertices) ofpolyhedral numbers.33 These weights, which Descartes denotes by O, can be obtained bycounting the number of polygonal faces opposite the solid angle chosen as origin (F),the number of edges (R, for “radix”), and the number of vertices (A, for “angles”). Usingthe formula for the gnomon, F � R + A, the total weight O can then be calculated [Des-cartes, 1987, pp. 38–39].

As one can see, this exercise is mainly a matter of counting, as is the first part of the textcopied by Leibniz, which deals with solid geometry and which contains (although not in themodern “Euler formula” form) the famous relationship between the number of faces, edges,and solid angles for regular polyhedra. It thus corresponds to the part of the mathesis thatDescartes ascribed to the consideration of “order” in the Regulae.34 These enumerations areconducted by recourse to a representation in which (polygonal) surfaces are reduced tomore basic figures (triangles) composed of lines, the extremities of which are the differentpondera. Hence they satisfy the requirement that proportion (we shall see that the term isused by Descartes in the text) be studied through the consideration of decomposition into“simple lines,” the extremities of which are points — even if in a meaning quite differentfrom that of the “calcul géométrique.” They also satisfy the requirement that these compu-tations, when involving complex configurations, be treated with the help of algebraic for-mulas (written at that time in cossic notation).

At the beginning of the passage on polyhedral numbers, for example, Descartes begins byrecalling how one can obtain the weight of any polygonal number by decomposing it intotriangles. He gives the example of the tretragonal weight, the number of which can be cal-culated as twice the number of triangular numbers minus the number of edges in com-mon.35 He then gives the general formula for regular polyhedral numbers in a similarmixed form of cossic notation and rhetorical discourse involving a description of the spatial

33 The first line of the second part of the text copied by Leibniz makes direct reference to this kind ofrepresentation: “Omnium optime formabuntur solida per gnomones superadditos uno semper angulovacuo existente, ac deinde totam figuram resolvi posse in triangula” [AT X, 269].34 To those who object that “order and measure” should be interpreted in a much broader meaning,grounding a technique which culminated in the “calcul géométrique” of 1637 or referring to themethodical “order,” I refer to a letter of Descartes to Ciermans, which has not yet received enoughattention from Cartesian scholars. Perceiving the generality of the technique involved inLa Géométrie, Ciermans proposed to call it a mathesis pura rather than a “geometry.” Descartesresponded negatively to this proposition by saying: “for I did explain in this treatise not onequestion pertaining properly to Arithmetic, nor even one of those concerned with order andmeasure, as did Diophantus; but, moreover, I didn’t deal in it with movement, even if puremathematics, at least the one which I have at most cultivated, has it for its principal object”[AT II, 70–71: my emphasis]. Note also the mysterious claim according to which the “principalobject” of pure mathematics is in 1638 movement. On this aspect see [De Buzon, 1996] and[Domski, 2009].35 “1

2 + 12 per 2 fit 2

2 + 22 unde sublato 1 fit 1 ” [Descartes, 1987, p. 4]. As one can see, the text

uses the astrological symbol for the unknown, a modification that is thought to come fromLeibniz. The general formula for polygonal numbers was already given by Roth, as were the ones forpyramidal numbers, which is used by Descartes in the last part of the passage without explanation[Schneider, 2008, p. 10].

448 D. Rabouin

configuration of these numbers.36 His main goal here seems to be to extend these results,already obtained by Faulhaber and Roth at the beginning of the century to semiregularpolyhedra, which he does in the last part of the text — the results of these different stepsbeing collected into several tables.

This kind of practice fits very well into the idea that an important part of mathematicsdeals with the consideration of order and that these computations can be performed withthe help of schematic spatial representations using points, lines, and figures (the basic figurehere being not the rectangle or the square, but the triangle). Moreover, in the De solidorumelementis, “order” is contrasted with “measure”, in accordance with what was written in theRegulae. This development reminds us very much of those we have already encounteredconcerning the notion of dimension in the Specimen and in the Regulae:

36 “Tthe fadifferradic37 Noejusda vertextsusualin thconsi

When we will imagine these same figures as measurable, then all the units will be under-stood as being of the same ratio as the figures themselves. For example, triangles [aremeasured by] triangular units; pentagons are measured by a pentagonal unit, etc. Thenthe proportion between a plane [figure] and its radix would be the same as that betweena square and its radix; and for a solid, [it will be] the same as that of a cube [to its radix]:for example, if the radix is 3, the plane will be 9, the solid 27, etc. This holds also for thecircle and the sphere, and all other figures. If the circumference of a circle is three timeslarger than another, its area will be nine times larger. From which it is observed that theprogressions involved in our mathesis, , , etc. are not tied to linear, square or cubicfigures, but designated in a general way through these diverse kind of measure. [Mytranslation of Descartes, 1987, pp. 4–5]

Even if this development is very intricate, the parallel with the Regulae seems no lessstriking: the first sentence repeats the idea (on the same examples of the triangle and thepentagon) that the notion of dimension can be seen as the number of parameters of theproblem (in this case the linear dimension of the given figure).37 This passage also echoesthe fact that the unit should have “as many dimensions as the extreme terms which areto be compared” [AT X, 450]. Furthermore, we now have a way to understand the apparentcontradiction between this last qualification and the identification of units with “points”: itamounts to the difference between the consideration of order (counting the number ofpondera, in which case the unit is a point) and of measure (measuring areas and volume,in which case the unit needs to have as many dimensions as the surface being considered).In the margin, Leibniz draws a picture (Fig. 7) that makes it easier to grasp what Descarteshas in mind [Descartes, 1987, p. 39].

The last sentence of the above quotation expresses the fact that already at the time of theDe solidorum elementis, Descartes had arrived at the idea, once again in perfect accordance

he algebraic expressions for these figurate numbers are found by multiplying the exponents ofce plus 1

2 by 13 + 1

3, and then by the number of faces, which is done as many times as there areent types of faces in the given body; then add to or subtract from the result the number ofes multiplied by 1

2 + 12 , and the number of angles multiplied by ” [Descartes, 1982, p. 108].

te that Descartes designates the identity of dimension by the expression “the same ratio”

em rationis, and that he also talks of the “proportion” between the plane figure and its “side” —y interesting choice of terminology if one wants to understand what he has in mind in otherwhen talking about “ratio and proportion” and that might well be much broader that what isly thought. This passage can be compared with the beginning of the development on dimensioe Regulae, in which it is defined as “modum et rationem, secundum quam aliquod subjectumderatur esse mensurabile” [AT X, 447, my emphasis].

Fig. 7. In the margin, Leibniz draws a triangle and a pentagon, whose side is three units and whosearea is nine units [from a facsimile of the Hanover manuscript, Descartes, 1987].


with the Regulae, that algebraic powers should not be interpreted in terms of a certain fixedgeometrical representation, but in terms of proportions entering into a certain relation (inthis case that of measure). This gives a very nice context for understanding how Descartescould at the same time have such clear ideas about the fact that algebraic degrees shouldnot be confused with geometric dimensions and nonetheless be apparently so unclear whenexpressing the notion of dimension in and of itself.

Even if there is no reason to conflate mathesis and mathesis universalis, the expression“the progressions involved in our mathesis” (“has progressiones notrae matheseos”) is alsovery striking. It indicates that Descartes thought that he had already elaborated a propermathesis in which the algebraic progressions played a central role — but independently ofthe exponential notation. It will not be necessary to insist on the fact that this mergingof algebra and geometry then had a meaning totally different from the one it would lateracquire. The reader should keep this in mind when reading Descartes’ claims that, in hisfirst mathematical studies at the beginning of the 1620s, he did “borrow all that is bestin Geometrical Analysis and Algebra, and correct the errors of the one by the other.”

In addition, the comparison between the Progymnasmata and the Regulae gives meaningto other intriguing passages in the latter, which reinforces the impression of proximitybetween the two texts. For example, in Rule X, Descartes states that: “In order to acquirediscernment we should exercise our intelligence by investigating what others have alreadydiscovered, and methodically survey even the most insignificant products of human skill,especially those which display or presuppose order” (my emphasis). This text is therefore aprivileged one in understanding what ordo meant for Descartes. Yet here are the examplesgiven:

[one] must first tackle the simplest and least exalted arts, and especially those in whichorder prevails — such as weaving and carpet-making, or the more feminine arts ofembroidery, in which threads are interwoven in an infinitely varied pattern. Number-games and any games involving arithmetic, and the like, belong here. (. . .) For, since noth-ing in these activities remains hidden and they are totally adapted to human cognitivecapacities, they present us in the most distinct way with innumerable instances of order,

38 Thinto acossis

450 D. Rabouin

each one different from the other, yet all regular. Human discernment consists almostentirely in the proper observance of such order. [AT X, 403–404; Descartes, 1985–1991, I, 34–35: my emphasis]

So at that time Descartes held an opinion regarding “Number-games and any gamesinvolving arithmetic” (compared with weaving, carpet-making, and embroidery) quite dif-ferent to the one he held later when he mocked mathematicians who wasted their time withsuch useless matters [AT II, 91, to Mersenne, 31 March 1638]. In the next passage in the texthe even gives the example of cryptography, an essential feature of the “number games” thatGerman Rechenmeister were playing in their “algebra” (Wortrechnung).38

I am not claiming, of course, that the De solidorum elementis provide the one and only keyto reading the Regulae. However, it is certainly of interest if one wants to understand some ofits aspects and it should be noted that the comparison has not previously been made by Carte-sian scholars. It has the tremendous advantage of relying on documents that are at the sametime both extant and compatible with the mathematical weaknesses contained in the Speci-men. If one follows this direction, how are we then to answer our second question?

3.4. Calculating with segments

As I emphasized in previous sections, Descartes had two different uses of geometricalschemas, which I have called “foundational” and “operative”. This designation is nottotally accurate since, as we have seen in the study of the De solidorum elementis, the “foun-dational” use is not solely a matter of justification, but can also support various kinds ofcomputation. My point is that, even in this case, the figuration does not represent the com-putation in and of itself. Neither in the Progymnasmata, nor in the Specimen, nor in Beeck-man’s Journal, nor even in Rule XIV of the Regulae do we find examples of the second usestrictly speaking. The schematism is merely an aid to solving certain problems, and this aidcan take various forms, including the use of dots and of basic figures other than squares andrectangles (typically triangles, into which polygons can be decomposed). What I have calledthe “operative” use seems to appear only in Rule XVIII, in which Descartes tries to repre-sent the basic operations between magnitudes in general. This coincides, however, with thesudden disappearance of points. How are we to understand this?

As should be clear now, my understanding is that Descartes is turning to another aspectof his project with Rule XVIII. But to distinguish these two aspects would not solve ourmain difficulty, that of the apparent incompatibility between them. It does not seem possibleto consider dots as represented by the extremities of segments without losing the very effi-ciency of the calculus (just think what happens to the extremities when two segments areadded). More generally — as pointed out by Stevin — the interpretation of units by pointscontradicts the very idea that an algebraic equation is a transcription of a series of magni-tudes in continuous proportion.

It is possible that what we face here is a mere contradiction. The Regulae is an unfinishedtreatise and it stops abruptly precisely after the exposition of the calculus in Rule XVIII, theremaining Rules (until XXI) subsisting only through their planned titles. It is not totallyimplausible that Descartes was not satisfied with this sketch, or that he saw that he was

ere is also an interesting reference in Rule VII to the game consisting of transforming its namen anagram [AT X, 391, 18]. This game played an important role in the fight between Germants; see [Mehl, 2001, pp. 192–204].

Fig. 8. A rectangle with a point in it? [Descartes, 1701, p. 57].


not taking it in the right direction. I do not think, however, that this type of explanation istotally satisfying, precisely because it leaves us with such a crude contradiction in Descartes’thought as the one mocked by Stevin. One should at least try to see if another interpretationis possible. And it turns out that it is. It rests, however, on a very small detail, which couldeasily escape our attention.

Recall that Descartes proposed in Rule XV to represent the unit in three ways: as asquare, as a line, or as a point. When one considers two magnitudes at a time, they canbe represented as a rectangle composed of unit squares or as a lattice of points, if theyare commensurable, or as an indecomposable rectangle, if they are incommensurable. Whenwe deal with only one of these magnitudes, we can picture it either as a line or a series ofdots, or as a rectangle. In this last case, one has just to take a rectangle whose side is thegiven magnitude and the other the unit. But in the picture reproduced in the Opuscula(see Fig. 8), the rectangle contains. . . a point • .

We have no way of knowing if this representation was Descartes’ original one, or a sug-gestion of the copyist, or even a misprint. But since it offers an alternative proposition tothe (disastrous) one consisting of interpreting dots as extremities of line segments, it shouldattract our interest. The passage is very intricate, but one thing at least seems to be clear:Descartes is fully aware of the fact that the representation through dots and that throughcontinuous magnitudes are not isomorphic. He divides the representation through rectan-gles into two cases according to whether or not we can divide the rectangle into unit cells. Inthe case of two incommensurable magnitudes, it is not possible to find a unit square intowhich their “rectangle” can be decomposed (since, by definition, there is no common mea-sure into which they can both be decomposed). The representation by dots is therefore seenas a subclass of the more general one through rectangles (and not an equivalent one, whichwould make us fall within Stevin’s critique).

The nature of the relationship between the two representations is not clear: on the onehand, Descartes insists on the fact that there are two different kinds of schematism depend-ing on whether or not we are dealing with multitudines; on the other hand, he proposes asingle schematism based on segments and rectangles without making clear whether onehas to interpret it in different ways depending on the objects under consideration, or if itis one among two different representations. But we have no reason to try to reduce this dif-ficulty, which is inherent to the tension between the foundational and the operative use ofthe schematism. What I want to emphasize is that there is, in any case, a way to escape Ste-vin’s critique and maintain the use of dots as “units.”

Here we should also recall the proposition made to Beeckman in the Specimen. When intro-ducing his schematism through unit squares and adding ita etiam punctum, Descartes did notdevelop a parallel representation through dots, but “by lines” involving “point, line, squareand cube.” We understand now how this very strange claim could nonetheless remain inaccordance with the idea that algebraic powers are magnitudes in continuous proportion.We just have to represent points “inside” our rectangle to render it consistent (see Fig. 9).

There is, however, a price to pay for this solution, since we now lose the close connectionwith the computation on dots (as extremities of lines) presented in the De solidorum elemen-

Fig. 9. Punctum, lineam, quadratum, cubum. Diagram adapted from a figure of the facsimile of fol.333 of Beeckman’s autograph published in (Beeckman, 1939–1953, III).

452 D. Rabouin

tis. My overall impression is that these two representations originate from two differentlines of thought, which do not match where they overlap.

3.5. The limitation of the geometrical calculus

As an indication of the fact that the representation of the calculus presented in Rule XVIIIis a sketch, which could correspond to a new line of thought for Descartes, I shall stress itslimitations. They appear clearly in the question of mean proportionals, at which point theextant part of the treatise abruptly stops. The first remark we should make in this regard isthat Descartes, at the time of the composition of the Regulae, isolated four basic operations,not five, as was usual at that time and as he would do in La Géométrie. This is due to the factthat root extraction is not seen as a basic operation but as a special case of division:

39 Thcruciameththereway.operahaveequatto awhich

As for those divisions in which the divisor is not given but only indicated by some rela-tion, as when we are required to extract the square root or the cube root etc., in thesecases we must note that the term to be divided, and all the other terms, are always tobe conceived as lines which form a series of continued proportionals, the first memberof which is the unit, and the last the magnitude to be divided. We shall explain in duecourse how to find any number of mean proportionals between the latter two magni-tudes. [AT X, 467; Descartes, 1985–1991, I, 75]

The last sentence is very striking. It means that at that time Descartes thought that he had ageneral procedure to insert any number of mean proportionals between two given magni-tudes. It could be a reference to his famous proportional compasses discovered in 1619, whichdo precisely this job. This has to remain a conjecture, since the last part of the treatise in whichDescartes was supposed to present this general procedure is not extant (lost or never writ-ten).39 In any case, Descartes has to explain first how to perform this operation in the simplecase of one mean proportional, an operation equivalent to the extraction of the square root.

e last Rules of the second part, of which we just have the titles, give us no clue about thisl issue. They just say that one must reduce any problem to an equation (XIX: “Using this

od of reasoning, we must try to find as many magnitudes, expressed in two different ways, asare unknown terms, which we treat as known in order to work out the problem in the directThat will give us as many comparisons between two equal terms”), and then carry out thetions (XX: “Once we have found the equations, we must carry out the operations which weleft aside, never using multiplication when division is in order”) and try to reduce a system ofions to a single one (XXI: “If there are many equations of this sort, they should all be reducedsingle one, viz. to the equation whose terms occupy fewer places in the series of magnitudes

are in continued proportion, i.e. the series in which the order of the terms is to be arranged”).

Fig. 10. Definition of the gnomon and the complements at the beginning of Euclid’s Elements II [inClavius, 1611–1612, Vol. 1, p. 83]. Just after this definition, Clavius presents the general procedure ofBook II as the foundation of the rules of algebra.


As we have seen, a key feature of Descartes’ calculus in the Regulae is that one works onsegments and on rectangles at the same time. This implies that one is able to transform anyrectangle into a line and vice versa. Since we already know how to represent any line by arectangle (whose sides are the given line and a unit), what we need is a general procedure totransform a given rectangle into another one. This is the general problem on which theRegulae abruptly stops:

It is therefore important to explain here how every rectangle can be transformed into a line,and conversely how a line or even a rectangle can be transformed into another rectangle,one side of which is specified. Geometers can do this very easily, provided they recognizethat in comparing lines with some rectangle (as we are now doing), we always conceivethe lines as rectangles, one side of which is the length which we adopted as our unit. In thisway, the entire business is reduced to the following problem: given a rectangle, to constructupon a given side another rectangle equal to it. The merest beginner in geometry is of courseperfectly familiar with this; nevertheless I want to make the point, in case it should seemthat I have omitted something. [AT X, 468; Descartes, 1985–1991, I, 76]

The rest is missing.“The merest beginner in geometry” would certainly know how to proceed in this case,

since it involves one of the basic procedures explained in Book II of Euclid’s Elementsand known as “application of areas.” The list of identities exposed in Book II is indeedbased on the fact that the “gnomon” constructed around the diagonal of a rectangle com-prises two equivalent rectangles called by Euclid the “complements” (see Fig. 10).

It is therefore easy to use this property and construct a rectangle equivalent to anotherone, for example, a “line” c (i.e., “line-rectangle”) equivalent to a “product” ab [Bos, 2009,see Fig. 11].

But our “merest beginner” would also know that there is a particular case in which thisoperation is not so easy to perform: when one of the sought-for rectangles is a square. In thiscase, which is equivalent to the insertion of a mean proportional, one needs, as Euclidexplained in the last proposition of Book II (II, 14: “to construct a square equivalent to a givenfigure”), the help of an auxiliary circle. But neither in the schematism presented in the Regulae,nor in the one presented to Beeckman in 1628, is there any mention of a circle. In contrast, theGéométrie would not only propose another schema for the basic operations on “simple lines,”

Fig. 11. How to construct a “line-rectangle,” whose side c is equal to the product ab, u being theunit.

454 D. Rabouin

but also would introduce a fifth basic operation, the extraction of a square root, which wouldbe represented, in the traditional Euclidean way, by resorting to a semicircle.

This discrepancy fits very well into the general context which I have tried to reconstructabove. Indeed, the representation of the extraction of square roots by a circle, which wasvery well known as a geometric procedure, does not have an immediate correspondent inthe representation of computation on multitudines. Thus it is no particular surprise thatDescartes simply “forgot” to include the circle as a basic element of his schematism in1628. Moreover, as mentioned earlier, Clavius used “application of area” in the part ofhis Algebra dedicated to the extraction of roots, but did so without resorting to the circleas shown in Fig. 12. This was because Clavius’s goal was not constructing the square root,but determining its value. He could therefore proceed by first drawing the sought-for squareand then determining the value of its side by using a gnomon property (in a way that is veryclose to the kind of technique that Descartes presented to Beeckman in his Specimen).

If this interpretation is right, it would have another important consequence for the studyof Cartesian mathematical texts of the late 1620s. It would mean that the inclusion of thesolutions of the third- and fourth-degree equations and of the Specimen into the samegroup of documents is far from obvious — the reason being that the first one was donethrough the intersection of a parabola and a circle.

3.6. Per parabolam

In February 1629, Beeckman transcribed in his Journal the content of new documentswhich he had received from Descartes, presumably after he himself had sent his demonstra-tion of the fact that the hyperbola was an anaclastic curve. These documents comprise first,after Beeckman’s demonstration, a solution to the problem of the insertion of two mean

Fig. 12. Extraction of square root in Clavius [1609, Chap. XIII, p. 52]. One first constructs thesquare ABHI and then determines the value of its side.


proportionals. This solution, which was achieved through the intersection of a circle and aparabola, was demonstrated geometrically by a “Parisian mathematician” (presumablyMydorge) after Descartes discovered its construction.40 The second text presents a generalmethod for constructing all solid problems — a “universal secret” to solve equationsinvolving three and four dimensions.41 We know from other documents that Descartes

40 [Beeckman, 1939–1953, Vol. IV, p. 156]. Cum D. des Chartes invenisset per parabolam duo mediaproportionalia inveniri, hoc mathematicus quidam Galus Parisiis geometrice demonstravit hoc modo.Beeckman then adds: quod ad verbum descripsi.41 Auxilio parabolae omnia solida problemata generali methodo construere. Quod alio loco vocatD. Des Chartes secretum universale ad aequationes omnes tertia vel quarta dimensione involutas lineisgeometricis exponendas [Beeckman, 1939–1953, Vol. IV, pp. 138–139]. Once again Beeckman writes,Quod ex illius scriptis ad verbum describo. Remember that Descartes promised in 1628 to sendBeeckman his Parisian Algebra. A passage from the letter of 1630 would indicate that he did so(Scribis enim Algebram, quam tibi dedi, meam amplius non esse [AT I, 159]). It could then be thoughtthat the solution of the third degree equation was part of this package. On the other hand, Descartesstates in the letter dated 25 January 1638 to Mersenne that nobody has a copy of his Algebra [AT I,501]. If this is true, it would mean that he just sent Beeckman an excerpt of it, perhaps his resolutionof the “solid problem,” as is suggested by the fact that his construction is presented just after thedemonstration by “a certain French mathematician.”

456 D. Rabouin

discovered his solution of the insertion of two mean proportionals in around 1625 [Bos,2001, p. 255; Serfati, 2002, pp. 72–75]. We do not know, however, how he discovered thegeneralization to the solution of any equation of degree 3 or 4 — for which, once again,he just gives the construction. Neither do we know if the “general” solution was discoveredbefore, after, or at the same time as the particular one.

Henk Bos has proposed an elegant reconstruction of the demonstration of the general-ization using the method of “indeterminate coefficients” [Bos, 2001, p. 257]. By just writingthe general equation of a circle and using the fact that the points should also lie on a para-bola, one immediately obtains a fourth-degree equation in one variable expressed only interms of the coordinates of the center of the circle. By then identifying these coefficientswith that of a general equation of degree 4, one obtains the different magnitudes involvedin Descartes’ construction. The method of “indeterminate coefficients” was known andused by cossist algebraists, but in an arithmetical context. By using it in a geometrical con-text in which curves are represented and manipulated through their equations, Descarteswould have made his first step in what would be the core of La Géométrie’s new technique.In fact, most commentators, when confronted with the text of 1629, simply replace it bythat of La Géométrie, in which the third part is dedicated to the solution of solid problem.42

If these reconstructions are correct, Descartes was certainly doing algebraic analysis in1628–1629. But unless new elements are brought into the debate, it would remain basedon a petitio principii (if Descartes was doing algebraic analysis, the reconstructions couldbe correct, and if they are, Descartes was doing algebraic analysis). As confessed by Bos,“it may be that Descartes arrived at the general construction by the method of indetermi-nate coefficients. The proof which I have added above shows that such a technique leadsdirectly to the construction. Moreover, in his notes to the 1659 Latin edition of the Geo-metry Van Schooten added a derivation of the construction by indeterminate coefficients.However, it may also be that Descartes found the general solution by successive generaliza-tion of his construction of two mean proportionals” [Bos, 2001, p. 258].

For my part, I would like to insist once again on the actual content of the text. Oneshould first note that this development is entitled (maybe by Beeckman) Parabolâ aequati-ones cossicas lineis exponere. As we have seen, Descartes does seem to have still been usingcossic notation at the time. The matter is one of importance, since many reconstructionswould not fit with this situation. For example, it is not easy to express the general equationof a circle using cossist symbols.43 The argument could, however, seem to be of little value,since in fact Beeckman’s transcription does not use symbolic notation at all. But the factthat the text, which is supposed to be transcribed ad verbum, is purely rhetorical shouldbe, in and of itself, very interesting. In Descartes’ description, the coefficients of the generalequation of fourth degree are described not by letters but by expressions such as “the num-ber of squares,” “the number of roots,” and “the absolute number,” in the same way as theywere described, for example, in Clavius [1609].

42 See for example [Serfati, 2002, pp. 95–104]. According to Serfati, Beeckman “s’est contenté detranscrire dans son langage cossique archaïque, sans pouvoir véritablement comprendre ni l’intérêtauthentique de la découverte cartésienne, ni surtout le système symbolique mathématique employépar son ami qui le lui exposait” (p. 101).43 Not easy, but not impossible considering the elements given in [Manders, 2006]. In Faulhaber’snotation, it was possible to introduce a second variable and to deal with indeterminate quantities —even if, to my knowledge, he did not do both at the same time.


But the main argument against this type of reconstruction is that it assumes thatDescartes, in 1628–1629, was studying curves through their equations. This claim isimmediately problematic with regard to the circle, since Descartes does not use thisequation even in later works, instead relying on the Pythagorean theorem [see Galuzziand Rovelli, forthcoming, Chap. 2.6]. With regard to the parabola, one has to keep in mindthat Descartes also sent Beeckman in 1628 some texts concerning conic sections, a topic hehad studied in the course of his works of optics (see Note 10). The main document concernsovals, which Descartes studied in an algebraic way in a fragment preserved in the Excerptamathematica [AT X, pp. 310–324] and in which he developed his first attempt at what wouldlater be known as the “method of normals” [Maronne, in this issue]. In this context, it isvery striking that the documents preserved by Beeckman present a purely geometricalanalysis with no use of algebraic techniques at all.

Finally, one should take into account the fact that the general context of this demon-stration is not the study of curves, but the theory of equations and more specifically oftheir “construction.”44 To reiterate the point made earlier: to argue that, with respectto the teleology leading towards the “calcul géométrique” of 1637 (i.e., the algebraicanalysis of curves), Descartes did not reach the achievements usually ascribed to him, doesnot amount to saying that he was ipso facto a lesser mathematician. With regard to hisAlgebra, in particular, the document sent to Beeckman shows that Descartes was alreadywell advanced in many aspects of the subject (the study of a “general equation”; thereduction of degree; the use of factorization, which seems necessary in any case to obtainthe magnitudes entering into his construction). The attention brought by scholars to theGerman cossists active at the beginning of the 17th century has shown how far theiralgebraic skills were developed, and how close their work is to what can be found inthe theory of equations provided by Descartes in the third book of La Géométrie[Manders, 2006].45 It is hence plausible that Descartes was already well advanced in thesetechniques at the beginning of 1620, and the documents that we have studied do notcontradict this hypothesis.

But, following the arguments given in the preceding sections, there are no cogent reasonsfor thinking that the Specimen was part of the same project (i.e., the Algebra, dealing withthe general theory of equations and their construction) and that all of these elements werepasted together in a coherent enterprise. On the contrary, these arguments show that thereare cogent reasons to think that they are not. This would confirm Bos’s judgement that thePappus problem, studied at the end of 1631 on Golius’ suggestion, was for Descartes “thecrucial catalyst; it provided him, in 1632, with a new ordered vision of the realm of geometryand it shaped his convictions about the structure and the proper methods of geometry”[Bos, 2001, p. 283: my emphasis]. In my opinion, it is at this moment that Descartes wentback to his 1619 project (on the classification of geometrical problems in analogy witharithmetical ones) and merged it with that of the Regulae (to treat all problems as equationsand to use a geometrical calculus to solve them).

44 On this theory, see [Bos, 1984].45 One should also keep in mind that when Mersenne asked Descartes to send his “old algebra” toMydorge, he replied that he would find a much better version of it in the third part of La Géométrie[AT I, 501].

458 D. Rabouin

4. Conclusion

The line of interpretation I have presented offers to my mind three important advanta-ges: first, it relies, as far as is possible, on contemporary documents without projecting ontothem a hidden practice for which we have no evidence, direct or indirect; second, it givesmeaning to many texts which remain in the retrospective view mysterious; finally, it allowsus to put these texts into a coherent narrative without jumping too quickly to the diagnosisof crude contradictions in Descartes’ thought.

On the other hand, as I have tried to show, to say that Descartes was not well advanced in1628 with regard to the teleology leading to the Géométrie does not amount to saying he wasa bad mathematician, persuaded that confused ideas and poor technique could ground agrandiose reform of “all human knowledge.” The De solidorum elementis provides impres-sive results for a young man who had not received a very advanced mathematical education,even if they appear in a context which is not the one we would recognize as the “modern”(“Cartesian”!) one. These results were, without doubt, a way of borrowing “all that is best inGeometrical Analysis and Algebra, and correct[ing] the errors of the one by the other.”

Acknowledgments

I thank Sébastien Maronne for his invaluable help, without which this paper would not exist, aswell as Henk Bos for providing me with his article on the Regulae. I also thank Niccolò Guicciardiniand June Barrow-Green for their careful reading and invaluable comments on this paper.

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